Properties

Label 1-967-967.723-r1-0-0
Degree $1$
Conductor $967$
Sign $0.989 + 0.141i$
Analytic cond. $103.918$
Root an. cond. $103.918$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.867 − 0.497i)2-s + (−0.165 + 0.986i)3-s + (0.505 − 0.862i)4-s + (0.209 + 0.977i)5-s + (0.347 + 0.937i)6-s + (−0.999 + 0.0260i)7-s + (0.00975 − 0.999i)8-s + (−0.945 − 0.325i)9-s + (0.668 + 0.744i)10-s + (0.710 + 0.703i)11-s + (0.767 + 0.641i)12-s + (0.964 + 0.263i)13-s + (−0.854 + 0.519i)14-s + (−0.998 + 0.0455i)15-s + (−0.488 − 0.872i)16-s + (−0.0292 − 0.999i)17-s + ⋯
L(s)  = 1  + (0.867 − 0.497i)2-s + (−0.165 + 0.986i)3-s + (0.505 − 0.862i)4-s + (0.209 + 0.977i)5-s + (0.347 + 0.937i)6-s + (−0.999 + 0.0260i)7-s + (0.00975 − 0.999i)8-s + (−0.945 − 0.325i)9-s + (0.668 + 0.744i)10-s + (0.710 + 0.703i)11-s + (0.767 + 0.641i)12-s + (0.964 + 0.263i)13-s + (−0.854 + 0.519i)14-s + (−0.998 + 0.0455i)15-s + (−0.488 − 0.872i)16-s + (−0.0292 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.989 + 0.141i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (723, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.989 + 0.141i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.515553966 + 0.2495128674i\)
\(L(\frac12)\) \(\approx\) \(3.515553966 + 0.2495128674i\)
\(L(1)\) \(\approx\) \(1.714580183 + 0.1048913128i\)
\(L(1)\) \(\approx\) \(1.714580183 + 0.1048913128i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.867 - 0.497i)T \)
3 \( 1 + (-0.165 + 0.986i)T \)
5 \( 1 + (0.209 + 0.977i)T \)
7 \( 1 + (-0.999 + 0.0260i)T \)
11 \( 1 + (0.710 + 0.703i)T \)
13 \( 1 + (0.964 + 0.263i)T \)
17 \( 1 + (-0.0292 - 0.999i)T \)
19 \( 1 + (0.791 - 0.610i)T \)
23 \( 1 + (0.371 - 0.928i)T \)
29 \( 1 + (0.107 + 0.994i)T \)
31 \( 1 + (0.571 - 0.820i)T \)
37 \( 1 + (0.0422 - 0.999i)T \)
41 \( 1 + (-0.203 - 0.979i)T \)
43 \( 1 + (0.120 + 0.992i)T \)
47 \( 1 + (-0.177 + 0.984i)T \)
53 \( 1 + (0.898 + 0.439i)T \)
59 \( 1 + (-0.628 - 0.777i)T \)
61 \( 1 + (0.746 - 0.665i)T \)
67 \( 1 + (-0.425 + 0.905i)T \)
71 \( 1 + (-0.864 + 0.502i)T \)
73 \( 1 + (0.898 - 0.439i)T \)
79 \( 1 + (-0.216 + 0.976i)T \)
83 \( 1 + (0.719 + 0.694i)T \)
89 \( 1 + (0.996 - 0.0844i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.67267592313436938689249060896, −20.85459709179740486683314697487, −19.88657764754583240576349217896, −19.41778154096941849509157895960, −18.30464444658969778711113528781, −17.27285330361763235582476142786, −16.7803852773699607786158308821, −16.10259440767054251479795374915, −15.24384289190779111055744754730, −14.00051848833652462934948077983, −13.35045220246645211142562254503, −13.11114103661266793639049999969, −11.98189710643102366429116126254, −11.71247211525057993529533485188, −10.31992064224083832011863313831, −8.9253522258522056884970550089, −8.38169133204447854680231509831, −7.46907287478440887738493486466, −6.24105917735067305875018737398, −6.10850266182740336127126720851, −5.15656167854424395183290197245, −3.795013584743337986194558045989, −3.17669619162966308326882202199, −1.75029238966182566278636989822, −0.83794185066026555825641334142, 0.74109842340404314629584386191, 2.38560705532630587582267928906, 3.12867007443435583409006127754, 3.82452102907586955237016762492, 4.71443044824173678997596337389, 5.78196385745769261951042787516, 6.51196518082494538601247022493, 7.15757603068974111381963669754, 9.118866296171561599774672032118, 9.57776163488735242957147844224, 10.4025071754808066563342432975, 11.13194525198856721057103924991, 11.7689025900195816529963492252, 12.76729291566282564209575396870, 13.83141716426368804537731242636, 14.31206115251739822118021779278, 15.197529236420174013052220511246, 15.85588819817142832152882104106, 16.45924306705683618694466615840, 17.7276705350005306030729837970, 18.58603416796518056783470406323, 19.43805084954337654631421774254, 20.2559934300068676662571954794, 20.83184975990193196108372448688, 21.81836428053861829292305603626

Graph of the $Z$-function along the critical line